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Series: CDSNS Colloquium

In the infinite-dimensional KAM theory, solving the homological equations is the one of the main parts. Generally, the coefficients of the homological equations are constants, by comparing the coefficients of the functions, it is easy to solve these equations. If the coefficients of homological equations depend on the angle variables, we call these equations as the variable coefficients homological equations. In this talk we will talk about how to solve these equations.

Series: School of Mathematics Colloquium

The Burau representation plays a key role in the classical theory of braid groups. When we let the complex parameter t take the value -1, we obtain a symplectic representation of the braid group known as the integral Burau representation. In this talk we will give a survey of results on braid congruence subgroups, that is, the preimages under the integral Burau representation of principal congruence subgroups of symplectic groups. Along the way, we will see the (perhaps surprising) appearance of braid congruence subgroups in a variety of other contexts, including knot theory, homotopy theory, number theory, and algebraic geometry.

Friday, December 7, 2018 - 15:00 ,
Location: Skiles 170 ,
Rafael de la Llave ,
School of Mathematics ,
Organizer: Rafael de la Llave

<p>Given a Hamiltonian system, normally hyperbolic invariant manifolds and their stable and unstable manifolds are important landmarks that organize the long term behaviour.</p>
<p>When the stable and unstable manifolds of a normally hyperbolic invarriant manifold intersect transversaly, there are homoclinic orbits that converge to the manifold both in the future and in the past. Actually, the orbits are asymptotic both in the future and in the past.</p><p>
One can construct approximate orbits of the system by chainging several of these homoclinic excursions.</p>
<p>A recent result with M. Gidea and T. M.-Seara shows that if we consider long enough such excursions, there is a true orbit that follows it. This can be considered as an extension of the classical shadowing theorem, that allows to handle some non-hyperbolic directions</p>

Series: Combinatorics Seminar

Series: Algebra Seminar

Classical Kazhdan's theorem for Riemann surfaces describes the limiting behavior of canonical (Arakelov) measures on finite covers in relation to the hyperbolic measure. I will present a generalized version of this theorem for metric graphs. (Joint work with Chenxi Wu.)

Wednesday, December 5, 2018 - 14:00 ,
Location: Skiles 006 ,
Agniva Roy ,
Georgia Tech ,
Organizer: Sudipta Kolay

<p>The talk will discuss a paper by Gompf and Miyazaki of the same name.
This paper introduces the notion of dualisable patterns, a technique
which is widely used in knot theory to produce knots with similar
properties. The primary objective of the paper is to first find a knot
which is not obviously ribbon, and then show that it is. It then goes on
to construct a related knot which is not ribbon. The talk will be aimed
at trying to unwrap the basic definitions and techniques used in this
paper, without going too much into the heavy technical details.</p>

Series: Analysis Seminar

A set $\Omega\subset \mathbb{R}^d$ is called spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Back in 1974 B. Fuglede conjectured that spectral sets could be characterized geometrically by their ability to tile the space by translations. Although since then the subject has been extensively studied, the precise connection between spectrality and tiling is still a mystery.>In the talk I will survey the subject and discuss some recent results, joint with Nir Lev, where we focus on the conjecture for convex polytopes.

Series: Algebra Seminar

One variation of the Waring problem is to ask for the shortest non-trivial equations of the form f_1^d + ... + f_r^d = 0, under various conditions on r, d and where f_j is a binary form. In this talk I'll limit myself to quadratic forms, and show all solutions for r=4 and d=3,4,5. I'll also give tools for you to find such equations on your own. The talk will touch on topics from algebra, analysis, number theory, combinatorics and algebraic geometry and name-check such notables as Euler, Sylvester and Ramanujan, but be basically self-contained. To whet your appetite: (x^2 + xy - y^2)^3 + (x^2 - xy - y^2)^3 = 2x^6 - 2y^6.

Series: Other Talks

Thesis defense:
Advisors: Turgay Uzer and Cristel Chandre
Summary:
Thirty years after the demonstration of
the production of high laser harmonics through nonlinear laser-gas
interaction, high harmonic generation (HHG) is being used to probe
molecular dynamics in real time and is realizing its
technological potential as a tabletop source of attosecond pulses in the
XUV to soft X-ray range. Despite experimental progress, theoretical
efforts have been stymied by the excessive computational cost of
first-principles simulations and the difficulty of
systematically deriving reduced models for the non-perturbative,
multiscale interaction of an intense laser pulse with a macroscopic gas
of atoms. In this thesis, we
investigate first-principles reduced models for HHG using
classical mechanics. On the microscopic level, we examine the
recollision process---the laser-driven collision of an ionized electron
with its parent ion---that drives HHG. Using nonlinear dynamics, we
elucidate the indispensable role played by the ionic
potential during recollisions in the strong-field limit. On the
macroscopic level, we show that the intense laser-gas interaction can be
cast as a classical field theory. Borrowing a technique from plasma
physics, we systematically derive a hierarchy of
reduced Hamiltonian models for the self-consistent interaction between
the laser and the atoms during pulse propagation. The reduced models
can accommodate either classical or quantum electron dynamics, and in
both cases, simulations over experimentally-relevant
propagation distances are feasible. We build a classical model based on
these simulations which agrees quantitatively with the quantum model
for the propagation of the dominant components of the laser field.
Subsequently, we use the classical model to trace
the coherent buildup of harmonic radiation to its origin in phase
space. In a simplified geometry, we show that the anomalously high
frequency radiation seen in simulations results from the delicate
interplay between electron trapping and higher energy recollisions
brought on by propagation effects.

Series: Geometry Topology Seminar

Weinstein cobordisms give a natural relationship on the set of Weinstein domains. Flexible Weinstein domains are minimal with respect to this relationship. In this talk, I will use these minimal domains to construct maximal Weinstein domains: any two high-dimensional Weinstein domains with the same topology are Weinstein subdomains of a maximal Weinstein domain also with the same topology. Using this construction, a wide range of new Weinstein domains can be produced, for example exotic cotangent bundles of spheres containing many different closed exact Lagrangians. On the other hand, I will explain how the same line of ideas can be used to prove restrictions on which categories can arise as the Fukaya categories of certain Weinstein domains.